Noise-induced patterns in semiconductor nanostructures and time-delayed feedback control

We study the constructive influence of noise upon the nonlinear dynamics of current density patterns in semiconductor nanostructures, and its control by time delayed feedback methods. In particular, we investigate noise-induced pattern formation in a double barrier resonant tunnelling diode described by a nonlinear reaction-diffusion model. For this purpose the parameters of the system are fixed at values below the Hopf bifurcation where the only stable state of the deterministic system is a spatially inhomogeneous "filamentary" steady state, and oscillating space-time patterns do not occur. We show that the addition of weak Gaussian white noise to the system gives rise to spatially inhomogeneous oscillations. As the noise intensity grows, the oscillations tend to become more and more spatially homogeneous, while simultaneously the temporal coherence of the oscillations decreases. We demonstrate that the application of a time delayed feedback loop, similar to that used in deterministic chaos control, allows one to control the temporal coherence and the time scales of the space-time patterns. Furthermore, with increasing control strength, a transition from spatially inhomogeneous, spiky oscillations to spatially homogeneous oscillations can be induced.